The Role of Visual Representations in Advanced Mathematical Problem Solving: An Examination of Expert-Novice Similarities and Differences

Expert mathematicians are contrasted with undergraduate students through a two-part analysis of the potential and actual use of visual representations in problem solving. In the first part, a classification task is used to indicate the extent to which visual representations are perceived as having potential utility for advanced mathematical problem solving. The analysis reveals that both experts and novices perceive visual representation use as a viable strategy. However, the two groups judge visual representations likely to be useful with different sets of problems. Novices generally indicate that visual representations would likely be useful mostly for geometry problems, whereas the experts indicate potential application to a wider variety of problems. In the second part, written solutions to problems and verbal protocols of problem-solving episodes are analyzed to determine the frequency, nature, and function of the visual representations actually used during problem solving. Experts construct visual representations more frequently than do novices and use them as dynamic objects to explore the problem space qualitatively, to develop a better understanding of the problem situation, and to guide their solution planning and enactment of problem-solving activity. In contrast, novices typically make little use of visual representations.     

Distinguishing Features of Mechanical and Human Problem-Solving

Over the years, research in mathematical problem-solving has examined expert/novice problem-solving performance on various types of problems and subjects. In particular, DeFranco examined two groups of Ph.D. mathematicians as they solved four mathematics problems and found that although all were content experts, only one group were problem-solving experts. Based on this study, this article posits the notion that one distinguishing feature between experts and novices is that experts tend to look for special features of a problem and use algorithms only as a “fail-safe” system while novices act like “machines” relying on algorithms to solve the problems. Why? The article explores the idea that novice problem solvers learned to solve problems the way they learned proof, that is, in a formal, abstract and mechanizable way. Beliefs about proof and the culture in which it is practiced help frame a mathematician's view of the discipline and ultimately impacts classroom practice. The authors believe that current classroom instruction tends to create a culture that fosters algorithmic proficiency and a “machine-like” approach to the learning of mathematics and problem-solving. Further, they argue that mathematicians need to be aware of the distinction between knowing a proof is true and explaining why it is true. When these distinctions are appreciated and practiced during classroom instruction, then and only then will students begin to acquire the mathematical knowledge to become better problem solvers.     

The role of problem representation and feature knowledge in algebraic equation-solving

Domain experts have two major advantages over novices with regard to problem solving: experts more accurately encode deep problem features (feature encoding) and demonstrate better conceptual understanding of critical problem features (feature knowledge). In the current study, we explore the relative contributions of encoding and knowledge of problem features (e.g., negative signs, the equals sign, variables) when beginning algebra students solve simple algebraic equations. Thirty-two students completed problems designed to measure feature encoding, feature knowledge and equation solving. Results indicate that though both feature encoding and feature knowledge were correlated with equation-solving success, only feature knowledge independently predicted success. These results have implications for the design of instruction in algebra, and suggest that helping students to develop feature knowledge within a meaningful conceptual context may improve both encoding and problem-solving performance.     

High school students' use of representations in physics problem solving

Findings from physics education research strongly point to the critical need for teachers’ use of multiple representations in their instructional practices such as pictures, diagrams, written explanations, and mathematical expressions to enhance students' problem‐solving ability. In this study, we explored use of problem‐solving tasks for generating multiple representations as a scaffolding strategy in a high school modeling physics class. Through problem‐solving cognitive interviews with students, we investigated how a group of students responded to the tasks and how their use of such strategies affected their problem‐solving performance and use of representations as compared to students who did not receive explicit, scaffolded guidance to generate representations in solving similar problems. Aggregated data on students' problem‐solving performance and use of representations were collected from a set of 14 mechanics problems and triangulated with cognitive interviews. A higher percentage of students from the scaffolding group constructed visual representations in their problem‐solving solutions, while their use of other representations and problem‐solving performance did not differ with that of the comparison group. In addition, interviews revealed that students did not think that writing down physics concepts was necessary despite being encouraged to do so as a support strategy.     

Understanding and representing the arithmetic averaging algorithm: an analysis and comparison of US and Chinese students' responses

Analysing the responses of 311 sixth-grade Chinese students and 232 sixth-grade US students to two problems involving arithmetic average, this study explored students' understanding and representation of the averaging algorithm from a cross-national perspective. Results of the study show that Chinese students were more successful than US students in obtaining correct numerical answers to each of the problems, but US and Chinese students had similar cognitive difficulties in solving the second task. The difficulties were not due to their lack of procedural knowledge of the averaging algorithm, rather due to their lack of conceptual understanding of the algorithm. There were significant differences between the US and Chinese students in their solution representations of the two average problems. Chinese students were more likely to use algebraic representations than US students; while US students were more likely to use pictorial or verbal representations. US and Chinese students' use of representations are related to their mathematical problem-solving performance. Students who used more advanced representations were better problem solvers. The findings of the study suggest that Chinese students' superior performance on the averaging problems is partly due to their use of advanced representations (e.g. algebraic).     

Problem solving in Chinese mathematics education: research and practice

This paper is an attempt to paint a picture of problem solving in Chinese mathematics education, where problem solving has been viewed both as an instructional goal and as an instructional approach. In discussing problem-solving research from four perspectives, it is found that the research in China has been much more content and experience-based than cognitive and empirical-based. We also describe several problem-solving activities in the Chinese classroom, including “one problem multiple solutions,” “multiple problems one solution,” and “one problem multiple changes.” Unfortunately, there are no empirical investigations that document the actual effectiveness and reasons for the effectiveness of those problem-solving activities. Nevertheless, these problem-solving activities should be useful references for helping students make sense of mathematics.     

Mathematical Thinking Involved in U.S. and Chinese Students' Solving of Process-Constrained and Process-Open Problems

This study examined U.S. and Chinese 6th-grade students' mathematical thinking and reasoning involved in solving 6 process-constrained and 6 process-open problems. The Chinese sample (from Guiyang, Guizhou) had a significantly higher mean score than the U.S. sample (from Milwaukee, Wisconsin) on the process-constrained tasks, but the sample of U.S. students had a significantly higher mean score than the sample of the Chinese students on the process-open tasks. A qualitative analysis of students' responses was conducted to understand the mathematical thinking and reasoning involved in solving these problems. The qualitative results indicate that the Chinese sample preferred to use routine algorithms and symbolic representations, whereas the U.S. sample preferred to use concrete visual representations. Such a qualitative analysis of students' responses provided insights into U.S. and Chinese students' mathematical thinking, thereby facilitating interpretation of the cross-national differences in solving the process-constrained and process-open problems.     

Mathematical Thinking Involved in U.S. and Chinese Students' Solving of Process-Constrained and Process-Open Problems

This study examined U.S. and Chinese 6th-grade students' mathematical thinking and reasoning involved in solving 6 process-constrained and 6 process-open problems. The Chinese sample (from Guiyang, Guizhou) had a significantly higher mean score than the U.S. sample (from Milwaukee, Wisconsin) on the process-constrained tasks, but the sample of U.S. students had a significantly higher mean score than the sample of the Chinese students on the process-open tasks. A qualitative analysis of students' responses was conducted to understand the mathematical thinking and reasoning involved in solving these problems. The qualitative results indicate that the Chinese sample preferred to use routine algorithms and symbolic representations, whereas the U.S. sample preferred to use concrete visual representations. Such a qualitative analysis of students' responses provided insights into U.S. and Chinese students' mathematical thinking, thereby facilitating interpretation of the cross-national differences in solving the process-constrained and process-open problems.     

Towards a System of Systems Methodologies

This paper sets out to consider O.R. as a problem-solving methodology in relation to other systems-based problem-solving methodologies. A ‘system of systems methodologies’ is developed as the interrelationship between different methodologies is examined along with their relative efficacy in solving problems in various real-world problem contexts. In a final section the conclusions and benefits which stem from the analysis are presented. The analysis points to the need for a co-ordinated research programme designed to deepen understanding of different problem contexts and the type of problem-solving methodology appropriate to each.     

How Students “Unpack” the Structure of a Word Problem: Graphic Representations and Problem Solving

This research investigated how fourth and fifth grade students spontaneously ‘unpacked’ a word problem when generating a graphic representation to aid in problem solution. Relationships among the type of graphic representation produced, spatial visualization, drawing ability, gender, and problem solving also were examined and described. Instrumentation developed for the study included several math challenge tasks, a spatial visualization task, and a drawing task. For one of the math challenge tasks, students were instructed to draw a picture to assist them with problem solution. These graphic representations generated by students were rated as pictorial or as displaying some level of schematic representation. Schematic representations included germane information from the problem supportive of problem solution. Pictorial representations included expressive and extraneous elements not necessary for problem solution, with no schematic elements. Findings indicated that the majority of students rendered schematic representations, with girls more likely than boys to use schematic representations at a statistically significant level. Students who used schematic visual representations were more successful problem solvers than those pictorially representing problem elements. The more “schematic‐like” the visual representation, the more successful students were at problem solution. Drawing a pictorial representation in the math challenge task also was negatively correlated to drawing skill.     

Identifying Fourth Graders' Understanding of Rational Number Representations: A Mixed Methods Approach

This study examined average‐, high‐ and top‐performing US fourth graders' rational number problem solving and their understanding of rational number representations. In phase one, all students completed a written test designed to tap their skills for multiplication, division and rational number word‐problem solving. In phase two, a subset of students sorted cards that showed part‐whole, ratio, quotient, measure, and operator perspectives of rational number representations. Each perspective was shown in numerical notational, word‐problem, and visual formats. The results indicated that top‐performing students scored significantly higher in problem solving and showed more effectively linked rational number representations than the other groups. The results imply that successful rational number problem solving is intertwined with representational knowledge for a wide range of rational numbers and that the bulk of US students do not possess effective skills for working with rational number representations.     

Psychological aspects of equation-based modelling

Psychological aspects of equation-based modelling languages like Modelica are under-represented in literature. This does not reflect the growth of the corresponding userbase. In this paper we try to close this gap by tackling the problem from three sides: we conduct expert interviews, we conduct an experiment based on self-reported timings to analyse the effects of inheritance on understandability, and we conduct an online experiment to analyse the effects of model representations on the performance at modelling tasks. The expert interviews suggest that experienced modelling experts tend to develop their models from the top-down, while novices do the opposite. Results from the second experiment indicate that the effect of inheritance on the time to understand a model is both significant and large. The results of the last experiment imply that graphical representations outperform block-diagrams for several metrics. These results open a broad research field on the theory of good modelling practice.     

Developing a Visual Interactive Model for Corporate Cash Management

The operational research/management science journals contain an extensive literature that addresses the corporate cash management problem; yet few, if any, companies make use of any of this published work in their daily cash-management decision making. A review of the literature suggests that the reason for this lack of applications may well be poor problem formulation—the problems that are solved in the literature as ‘cash management’ problems evolve from a ‘hard systems’ view of real-world cash management. However, the problem as perceived by cash managers involves both dynamic and loosely structured components which are difficult to model using classical (i.e. ‘hard systems’) approaches.We therefore decided to approach the cash management problem as an experiment in the use of a novel visual interactive problem solving (VIPS) methodology. The aim of the experiment was to develop an implementable, visual interactive model to support daily cash management decision making. Working closely with a corporate cash manager, we first developed a visual model of his daily decision problem and then agreed on the feasible options and the interactive requirements. At this stage, the problem was sufficiently well defined for a mathematical model to be built and the visual model made ‘smart’.This paper discusses the results of this experiment and suggests that VIPS may have distinct advantages as a problem-solving technique in loosely structured, ‘messy’ problem situations.     

Why do U.S. and Chinese students think differently in mathematical problem solving?: Impact of early algebra learning and teachers’ beliefs

This paper reports two studies that examined the impact of early algebra learning and teachers’ beliefs on U.S. and Chinese students’ thinking. The first study examined the extent to which U.S. and Chinese students’ selection of solution strategies and representations is related to their opportunity to learn algebra. The second study examined the impact of teachers’ beliefs on their students’ thinking through analyzing U.S. and Chinese teachers’ scoring of student responses. The results of the first study showed that, for the U.S. sample, students who have formally learned algebraic concepts are as likely to use visual representations as those who have not formally learned algebraic concepts in their problem solving. For the Chinese sample, students rarely used visual representations whether or not they had formally learned algebraic concepts. The findings of the second study clearly showed that U.S. and Chinese teachers view students’ responses involving concrete strategies and visual representations differently. Moreover, although both U.S. and Chinese teachers value responses involving more generalized strategies and symbolic representations equally high, Chinese teachers expect 6th graders to use the generalized strategies to solve problems while U.S. teachers do not. The research reported in this paper contributed to our understanding of the differences between U.S. and Chinese students’ mathematical thinking. This research also established the feasibility of using teachers’ scoring of student responses as an alternative and effective way of examining teachers’ beliefs.     

Using Metaphors to Understand and Solve Arithmetic Problems: Novices and Experts Working With Negative Numbers

In this study, novices and experts used the same metaphors to understand and solve problems with negative numbers. However, they used them differently. Twenty-four participants (12 middle school children and 12 postsecondary adults) computed arithmetic expressions during the problem-solving task. During this task, children used metaphors more often than adults did to compute, detect and correct errors, and justify their answers. Metaphorical computations were more accurate but slower than other methods. The participants explained 6 arithmetic expressions during the understanding task. During this task, the adults used more metaphors (with fewer details) and used them more often than the children did. Compared to the median child, the median adult showed a more integrated understanding of arithmetic through multiple metaphors, mathematical rules, and transformations. These results suggest that the metaphors used by both the children and the adults are central to understanding arithmetic. Thus, these metaphors are likely candidates for theory-constitutive metaphors.     

Using Metaphors to Understand and Solve Arithmetic Problems: Novices and Experts Working With Negative Numbers

In this study, novices and experts used the same metaphors to understand and solve problems with negative numbers. However, they used them differently. Twenty-four participants (12 middle school children and 12 postsecondary adults) computed arithmetic expressions during the problem-solving task. During this task, children used metaphors more often than adults did to compute, detect and correct errors, and justify their answers. Metaphorical computations were more accurate but slower than other methods. The participants explained 6 arithmetic expressions during the understanding task. During this task, the adults used more metaphors (with fewer details) and used them more often than the children did. Compared to the median child, the median adult showed a more integrated understanding of arithmetic through multiple metaphors, mathematical rules, and transformations. These results suggest that the metaphors used by both the children and the adults are central to understanding arithmetic. Thus, these metaphors are likely candidates for theory-constitutive metaphors.     

Learning to represent,representing to learn

This study explores how students learn to create, discuss, and reason with representations to solve problems. A summer school algebra class for seventh and eighth graders provided opportunities for students to create and use representations as problem-solving tools. This case study follows the learning trajectories of three boys. Two of the three boys had been low-achievers in their previous math classes, and one was a high achiever. Analysis of all three boys’ written work reveals how their representations became more sophisticated over time. Their small group interactions while problem-solving also show changes in how they communicated and reasoned with representations. For these boys, representation functioned as a learning practice. Through constructing and reasoning with representations, the boys were able to engage in generalizing and justifying claims, discuss quadratic growth, and collaborate and persist in problem-solving. Negotiating different student-constructed representations of a problem also gave them opportunities to act with agency, as they made choices and judgments about the validity of the different perspectives. These findings have implications for the importance of giving all students access to mathematics through representations, with representational thinking serving as a central disciplinary practice and as a learning practice that supports further mathematics learning.